GRADE 12 EXAMINATION NOVEMBER 2018

GRADE 12 EXAMINATION NOVEMBER 2018

ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA

Time: 2 hours

200 marks

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This question paper consists of 9 pages and an Information Booklet of 4 pages (i?iv). Please check that your question paper is complete.

2. Non-programmable and non-graphical calculators may be used, unless otherwise indicated.

3. All necessary calculations must be clearly shown and writing should be legible.

4. Diagrams have not been drawn to scale.

5. Round off your answers to two decimal digits, unless otherwise indicated.

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GRADE 12 EXAMINATION: ADVANCED PROGRAMME MATHEMATICS: PAPER I

QUESTION 1 1.1 Solve for x without using a calculator and showing all working:

(a) x2 + x =-2x - 2

(b) ln x3 + 2ln x2 = 7

Page 2 of 9

(8) (6)

1.2 The equation for radioactive decay of a radioactive element is:

=y y0e-kt , k > 0

where y0 is the initial amount, y is the amount after time t (in years) and k is a constant.

The half-life of an element is the time taken for half of the quantity to decay.

(a) Make k the subject of the formula.

(4)

(b) Determine the value of k for Carbon-14 if the half-life of Carbon-14 is

5 700 years. Give the answer correct to 6 decimal places.

(2)

(c) How old is a sample in which 10% of the Carbon-14 nuclei originally

present have decayed? In other words, 90% of the original quantity

remain.

(4)

[24]

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GRADE 12 EXAMINATION: ADVANCED PROGRAMME MATHEMATICS: PAPER I

QUESTION 2

Page 3 of 9

( ) 2.1 Determine, in standard form ax3 + bx2 + cx + d =0 with a, b, c and d real,

a cubic equation which has roots -3 and 3 + 2i.

(8)

2.2 Explain why every cubic equation with real coefficients must have at least

one real root.

(4)

2.3 Thabo is practising his division of complex numbers of the form a + bi

where a,b .

He notices that 3 + 2i = -i 5 - 7i = -i and

-2 + 3i

7 + 5i

4 + 5i = -i . -5 + 4i

Prove that a + bi = -i for all a,b .

(8)

-b + ai

[20]

QUESTION 3

Use Mathematical Induction to prove that 23n - 3n is divisible by 5 for n . [14]

QUESTION 4

4.1 Consider the function f ( x ) = e x

(a) Draw a sketch graph of f, showing at least 2 points on the graph.

(8)

(b) Write down the x-coordinate of a point at which f is not differentiable. (2)

4.2 Consider the function f(x), where a and b are rational.

f

(

x

)

=

ax - b - 1, bx2 - ax + 5,

x ................
................

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